# Tropical Geometry of Deep Neural Networks

arXiv: https://arxiv.org/abs/1805.07091 · HTML: https://mlsanity.com/paper/1805.07091/

## Key facts

- Cited by 95 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 May 2018.

## Built on by (most cited first, 30 of 95)

- [Towards Lower Bounds on the Depth of ReLU Neural Networks](https://mlsanity.com/paper/2105.14835/) (2021)
- [When Deep Learning Meets Polyhedral Theory: A Survey](https://mlsanity.com/paper/2305.00241/) (2023)
- [Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sums](https://mlsanity.com/paper/2104.08135/) (2021)
- [Lower Bounds on the Depth of Integral ReLU Neural Networks via Lattice Polytopes](https://mlsanity.com/paper/2302.12553/) (2023)
- [The Real Tropical Geometry of Neural Networks](https://mlsanity.com/paper/2403.11871/) (2024)
- [An enriched category theory of language: from syntax to semantics](https://mlsanity.com/paper/2106.07890/) (2021)
- [Algorithmic Determination of the Combinatorial Structure of the Linear Regions of ReLU Neural Networks](https://mlsanity.com/paper/2207.07696/) (2022)
- [Functional dimension of feedforward ReLU neural networks](https://mlsanity.com/paper/2209.04036/) (2022)
- [Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometry](https://mlsanity.com/paper/2501.18915/) (2025)
- [Neural Networks are Decision Trees](https://mlsanity.com/paper/2210.05189/) (2022)
- [Mildly Overparameterized ReLU Networks Have a Favorable Loss Landscape](https://mlsanity.com/paper/2305.19510/) (2023)
- [On the Expected Complexity of Maxout Networks](https://mlsanity.com/paper/2107.00379/) (2021)
- [Training Fully Connected Neural Networks is existsℝ-Complete](https://mlsanity.com/paper/2204.01368/) (2022)
- [Activation degree thresholds and expressiveness of polynomial neural networks](https://mlsanity.com/paper/2408.04569/) (2024)
- [Better Neural Network Expressivity: Subdividing the Simplex](https://mlsanity.com/paper/2505.14338/) (2025)
- [Neurons on Amoebae](https://mlsanity.com/paper/2106.03695/) (2021)
- [Exact Count of Boundary Pieces of ReLU Classifiers: Towards the Proper Complexity Measure for Classification](https://mlsanity.com/paper/2306.08805/) (2023)
- [On Minimal Depth in Neural Networks](https://mlsanity.com/paper/2402.15315/) (2024)
- [Depth-Bounds for Neural Networks via the Braid Arrangement](https://mlsanity.com/paper/2502.09324/) (2025)
- [On the Depth of Monotone ReLU Neural Networks and ICNNs](https://mlsanity.com/paper/2505.06169/) (2025)
- [Origami in N dimensions: How feed-forward networks manufacture linear separability](https://mlsanity.com/paper/2203.11355/) (2022)
- [Lower and Upper Bounds for Numbers of Linear Regions of Graph Convolutional Networks](https://mlsanity.com/paper/2206.00228/) (2022)
- [Abstract Visual Reasoning: An Algebraic Approach for Solving Raven's Progressive Matrices](https://mlsanity.com/paper/2303.11730/) (2023)
- [Decomposition Polyhedra of Piecewise Linear Functions](https://mlsanity.com/paper/2410.04907/) (2024)
- [Neural Networks and (Virtual) Extended Formulations](https://mlsanity.com/paper/2411.03006/) (2024)
- [On the Local Complexity of Linear Regions in Deep ReLU Networks](https://mlsanity.com/paper/2412.18283/) (2024)
- [Min-Max-Plus Neural Networks](https://mlsanity.com/paper/2102.06358/) (2021)
- [Noncompact uniform universal approximation](https://mlsanity.com/paper/2308.03812/) (2023)
- [The Resurrection of the ReLU](https://mlsanity.com/paper/2505.22074/) (2025)
- [Most ReLU Networks Admit Identifiable Parameters](https://mlsanity.com/paper/2605.03601/) (2026)

## Abstract (arXiv)

We establish, for the first time, connections between feedforward neural networks with ReLU activation and tropical geometry --- we show that the family of such neural networks is equivalent to the family of tropical rational maps. Among other things, we deduce that feedforward ReLU neural networks with one hidden layer can be characterized by zonotopes, which serve as building blocks for deeper networks; we relate decision boundaries of such neural networks to tropical hypersurfaces, a major object of study in tropical geometry; and we prove that linear regions of such neural networks correspond to vertices of polytopes associated with tropical rational functions. An insight from our tropical formulation is that a deeper network is exponentially more expressive than a shallow network.